Fano variety (original) (raw)
In der algebraischen Geometrie, einem Teilgebiet der Mathematik, ist eine Fano-Varietät eine vollständige Varietät über einem Körper , deren ist. Eine Fano-Mannigfaltigkeit ist eine singularitäten-freie komplexe Fano-Varietät.
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| dbo:abstract | In der algebraischen Geometrie, einem Teilgebiet der Mathematik, ist eine Fano-Varietät eine vollständige Varietät über einem Körper , deren ist. Eine Fano-Mannigfaltigkeit ist eine singularitäten-freie komplexe Fano-Varietät. (de) In algebraic geometry, a Fano variety, introduced by Gino Fano in (Fano , ), is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties. (en) 대수기하학에서 파노 다양체(영어: Fano variety)는 사영 공간과 유사하게, 반표준 인자가 풍부한 인자를 이루는 대수다양체이다. (ko) 代数幾何学では、ファノ多様体(Fano variety)は、( Fano , ) により導入され、多様体上の反標準バンドルが豊富な(complete variety) X のことを言う。この定義は、X がある定義体上で(smooth)なことを前提としているが、極小モデルプログラムでは、端末特異点(canonical singularity)やklt特異点(klt singularity)（川又対数端末特異点）といった、様々なタイプの特異点を持ったファノ多様体の研究も進められていた。 (ja) In algebraïsche meetkunde is een Fano-variëteit, geïntroduceerd door Gino Fano, een niet-singuliere complete variëteit waarvan de anti-kanonieke bundel een is. In het bijzonder hebben alle Fano-variëteiten een Kodaira-dimensie −∞. Fano-variëteiten in dimensies 1 zijn isomorf met de projectieve lijn. In dimensie 2 zijn Fano-variëteiten del Pezzo-oppervlakken en zijn zij isomorf ofwel aan of aan het projectieve vlak opgeblazen op hoogstens 8 algemene punten, en in het bijzonder zijn zij allen rationaal. In dimensie 3 zijn er niet-rationale voorbeelden. Iskovskih classificeerde de Fano 3-folds waar het tweede Betti-getal gelijk is aan 1 in 18 klassen, en Mori deed in 1981 hetzelfde voor Fano 3-folds met het tweede Betti-getal gelijk aan 2, waarbij hij 88 vervormingsklassen vond. (nl) |
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| dbp:authorlink | Gino Fano (en) |
| dbp:bot | InternetArchiveBot (en) |
| dbp:date | August 2019 (en) |
| dbp:first | Vik.S. (en) |
| dbp:fixAttempted | yes (en) |
| dbp:last | Fano (en) Kulikov (en) Iskovskih (en) |
| dbp:title | Fano_variety (en) |
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| rdfs:comment | In der algebraischen Geometrie, einem Teilgebiet der Mathematik, ist eine Fano-Varietät eine vollständige Varietät über einem Körper , deren ist. Eine Fano-Mannigfaltigkeit ist eine singularitäten-freie komplexe Fano-Varietät. (de) In algebraic geometry, a Fano variety, introduced by Gino Fano in (Fano , ), is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties. (en) 대수기하학에서 파노 다양체(영어: Fano variety)는 사영 공간과 유사하게, 반표준 인자가 풍부한 인자를 이루는 대수다양체이다. (ko) 代数幾何学では、ファノ多様体(Fano variety)は、( Fano , ) により導入され、多様体上の反標準バンドルが豊富な(complete variety) X のことを言う。この定義は、X がある定義体上で(smooth)なことを前提としているが、極小モデルプログラムでは、端末特異点(canonical singularity)やklt特異点(klt singularity)（川又対数端末特異点）といった、様々なタイプの特異点を持ったファノ多様体の研究も進められていた。 (ja) In algebraïsche meetkunde is een Fano-variëteit, geïntroduceerd door Gino Fano, een niet-singuliere complete variëteit waarvan de anti-kanonieke bundel een is. In het bijzonder hebben alle Fano-variëteiten een Kodaira-dimensie −∞. (nl) |
| rdfs:label | Fano-Varietät (de) Fano variety (en) ファノ多様体 (ja) 파노 다양체 (ko) Fano-variëteit (nl) |
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